Master 2 Pure and Applied Mathematics - 1st trimester

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  • Schedule: pdf.
  • Courses in the first trimester 2017-18
    • Bloc Algebra and Geometry
      • Complex Geometry (S. Dumitrescu, A. Höring - 6 ECTS) Final exam: November 30, 9h-12h, salle 2
        • Complex varieties (smooth or singular) are very rich geometric objects that can be studied from both the analytical and algebraic point of view. We will first introduce complex varieties of arbitrary dimension as well as the basic tools needed for their study. In the second part of the course we study in more detail the case of curves. This course provides the prerequisites for the courses of F. Labourie and Dimca-Parusinski. Topics of the first part are holomorphic functions in several variables, analytic sets in C^n, complex manifolds, vector bundles, differential forms of type (p, q), Dolbeault and de Rham cohomology, Riemann-Roch theorem on a compact Riemann surface. Topics of the second part are definition of Riemann surfaces, basic examples, associated algebraic curves and Riemann surfaces, Riemann surfaces associated with an analytic function, Riemann surfaces obtained as quotients by a group of discrete automorphismes, Riemannian metrics on surfaces, conformal transformations, hyperbolic geometry.
      • Local systems and monodromy (C. Cazanave, I. Waschkies - 6 ECTSFinal exam: December 6, 14h-17h, salle 2
        • The purpose of this course is to show how the fundamental group appears in the study of the behavior of solutions of a system of PDEs (linear). In the first part of this course are introduced  the concept of monodromy in algebraic topology - from a covering one constructs a representation of the fundamental group. We consider a sheaf perspective that is not only suitable for the study of PDEs (D-modules), but also in various generalizations of the notion of fundamental group in modern algebraic geometry (e.g. Galois theory in the sense of Grothendieck). Some of the topics covered in this course are fundamental group and coverings, monodromy representation, local systems and sheaves, homological algebra and sheaf cohomology. Time permitting we will study concrete examples of local systems from the cohomology of the complex solutions of a system of linear PDEs.
    • Bloc Analysis
      • Continuum mechanics (D. Clamond - 6 ECTSFinal exam: November 28, 14h-17h, salle 2
        • We introduce the mathematical techniques needed to describe continuous physical environments (tensors, PDEs, integral calculus, holomorphic functions, Fourier analysis, etc) in order to model continuous media. We then discuss applications to elastic solids and in particular to fluids (perfect or viscous, incompressible or not, homogeneous or not). The study of flows of fluids poses many difficulties, mathematics and physical: non-linear equations, non-uniqueness of solutions, instabilities, etc. Understanding these physics requires fine mathematical analysis and numerical modeling, which are addressed in the other parts of the course.
      • Functional analysis (P. Dreyfuss - 6 ECTSFinal exam: December 7, 9h-12h, salle 2
        • Navier-Stokes equations are a model, well accepted by engineers and physicists, which describes the flow of a fluid. The analysis of these equations is of great interest because it provides insight into the algorithms used to simulate the flow of fluids. In this introductory course we will study the basis of functional analysis bases that allow us to analyze the Navier-Stokes equations. Some of the topics covered are fundamental results on Sobolev spaces, especially those appearing in hydrodynamics, analytic techniques for elliptic and parabolic PDEs.
      • Evolution equations: from local to global (F. Planchon - 6 ECTSFinal exam: November 29, 9h-12h, salle 2
        • This course is an introduction to various tools that have become classics in the analysis of partial differential equations: interpolation real / complex,  functional spaces beyond the Sobolev spaces and Lesbegue, methods of (semi) groups and local Cauchy theory for simple models (parabolic fluid models, wave equations or semilinear Schrodinger, in the entire space or on a torus). Later we will discuss asymptotic questions by presenting some simple cases (such as the viriel methods on dispersive models).
    • Bloc Probability and Statistics (joint lectures with the Master MathMods)
      • Stochastic calculus for mathematical finance (C. Bernardin - 6 ECTS)
        • This course is devoted to the introduction of the basic concepts used in mathematical finance. It will consist of a presentation of Brownian motion, Itô integral, stochastic differential equations and Girsanov theorem. From the modelling point of view, all these tools will be used to introduce the notions of strategy, arbitrage and risk-neutral probability measure and to define the Black Scholes model used for the pricing of European options. A good knowledge in probability theory (with measure theory) is required.
      • Advanced statistics and applications (Y. Baraud - 6 ECTS)
        • The regression framework is a powerful modelling tool for understanding how a quantitative variable (the stock price,the income of a firm, the sales of a product…) varies as time goes by or depends on a set predictors. The aim of this course is to present some mathematical tools for statistical inference in this setting. We shall deal with the problems of non-parametric estimation, variable selection and testing the goodness of fit of a model, among others. Our point of view will mainly be non-asymptotic and based on a model selection approach. For illustration, we shall provide an application to finance and the problem of estimating the drift in the Black-Scholes model where this parameter is allowed to depend on time.
      • Numerical methods in probability for mathematical finance (E. Tanré and J. Inglis, INRIA - 6 ECTS)
        • Probabilistic numerical methods are widely used in mathematical finance for pricing financial derivatives and computing strategies. The course will present the basic methods used for simulating random variables and implementing the Monte-Carlo method. Simulation of stochastic processes used in mathematical finance, such as Brownian motion and solutions to stochastic differential equations, will be discussed as well.